Enhancing interpolation and approximation error estimates using a novel Taylor-like formula

In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the P₁-interpolation error estimate. Following the same principles, we have also developed a novel numerical scheme for the heat equation that provides a better error estimate compared to the classical implicit finite differences scheme. Numerical illustrations confirm this behavior.